Methodology for estimating statistical distribution characteristics of physical parameters of semiconductor device

ABSTRACT

A method for estimating statistical distribution characteristics of physical parameters of a semiconductor device includes manufacturing a plurality of semiconductor device chips, each having a plurality of transistors, preparing electrical characteristic data by measuring electrical characteristics of the plurality of transistors included in the plurality of chips, extracting an inter-chip distribution characteristic and an intra-chip distribution characteristic of the electrical characteristics by analyzing the electrical characteristic data, generating random number data satisfying the extracted inter-chip and intra-chip distribution characteristics, and performing a simulation for extracting statistical distribution characteristic data of the physical parameters of the chips, based on the random number data.

CROSS-REFERENCE TO RELATED APPLICATIONS

This U.S. non-provisional patent application claims priority under 35U.S.C. § 119 of Korean Patent Application No. 10-2006-0013189, filed onFeb. 10, 2006, the content of which is hereby incorporated by reference,in its entirety.

BACKGROUND OF THE INVENTION

Embodiments of the present invention disclosed herein relate to amethodology for estimating statistical distribution characteristics ofparameters for product development.

The quality of a semiconductor-device-based product is generallydependent upon design rules and process conditions which are applied tothe design and manufacture of the product. With the trend toward furtherintegration of such devices, the design and manufacturing processes ofindustrial products become increasingly complicated. As a result, it isincreasingly difficult to analyze the relationship of product qualityrelative to the design rules and the process conditions. Therefore,methods are required that can accurately and rapidly analyzecorrelations between the design rules or the process conditions and theproduct quality since enhanced accuracy and speed in analysis makes itpossible to reduce the time-to-market of a new product.

Specifically, in manufacturing a high-technology semiconductorintegrated circuit, the design and manufacturing procedures are quitecomplicated so that it is difficult to analyze the correspondingcorrelation. A manufacturer of a semiconductor integrated circuitfabricates the semiconductor integrated circuit based on a specificationthat defines requirements for electrical and structural characteristics.Early-on in the semiconductor industry, verification of a circuit designin accordance with a specification was performed by a human. However, assemiconductor circuits became more highly integrated, such verificationwas performed using a computer. Unfortunately, although computers havingremarkably excellent computing power are employed for this task, thespeed and accuracy of the circuit design verification is remarkablyreduced as semiconductor circuit technology becomes more highlyintegrated.

In addition, as semiconductor devices become further reduced in size, arelative rate of a process variation occurring during the manufacturingprocess of the semiconductor device becomes increased. That is, avariation rate of process error of design features of the same size withrespect to a reference size is further increased in a more highlyintegrated semiconductor integrated circuit. As a result, it isnecessary to consider process variation in the design of thesemiconductor integrated circuit. In particular, since the processvariation has a great effect on the yield of the semiconductor device,it is increasingly important to estimate variation of the electricalcharacteristics of the product in accordance with process variationduring the design stage.

Specifically, since the electrical characteristics of a semiconductordevice are dependent on structural/physical parameters (hereinafter,referred to as independent parameters) such as channel length (L),device width (W), doping profile (Na or Nd), oxide thickness (t_(ox)),oxide permittivity (ε_(ox)), channel length modulation constant (λ), orthe like, it is necessary to estimate the statistical distribution ofthe independent parameters in order to enhance the yield of thesemiconductor device. In a conventional method for estimating thestatistical distribution of the independent parameters, referring toFIG. 1, a predetermined simulation (S2) is performed to estimate theproduct characteristic (S3). For simulation, design data, i.e., theindependent parameters, are used as input data, wherein it is assumedthat the design data have a predetermined distribution characteristic,e.g., a normal distribution characteristic. However, due tocomplications, such as the process variation or the like, it may not beproper to assume that the input data has the normal distributioncharacteristic. Since an improper input data incurs an inadequateestimation for product characteristic, it is insufficient that thedesign data to be used as the input data is assumed to have the normaldistribution characteristic. Therefore, it should be necessary toestimate it properly.

Nevertheless, estimation of the statistical distribution of theindependent parameters is generally not straightforward. For instance,although it is possible to derive an equation expressing the correlationbetween the independent parameters and the electrical characteristicsdependent thereupon through physical theory, this approach is successfulonly in a very limited case. That is, in general, the equation may be amultivariable function, and further, variables of the equation aredependent upon process conditions, which are continuously updated forimproving the yield of the product. Therefore, in practice, it is quitedifficult to derive the equation through a theoretical approach. As aresult, it is also difficult to obtain an adequate estimation ofstatistical distribution of the independent parameters using theconventional method.

In addition, though the statistical distributions of the independentparameters may be obtained from an actual measurement for theindependent parameters in principle, it is impossible to measure them,in practice, because the time for measurement is too great. In order toovercome such a technical difficulty, another conventional method ofestimating the statistical distribution by modeling one of theindependent parameters can be employed. However, this conventionalmethod is still limited in that it cannot extract information aboutother, non-selected, independent parameters. In particular, since thesemodeling methods are based on a modeling fitting which requires a longprocedure for calculation, they cannot provide a physical meaning forthe correlation between the independent parameters and the electricalcharacteristics dependent thereupon. In addition, a large amount of timeis required for such a calculation.

SUMMARY OF THE INVENTION

Embodiments of the present invention provide a method for accurately andrapidly analyzing a correlation between independent parametersdetermining characteristics of products, and dependent parameters.

Embodiments of the present invention also provide a method forestimating statistical distribution characteristics capable ofunderstanding a physical relation between the independent parameters anddependent parameters, for improving product quality and reducingdevelopment period.

In one aspect, methods of estimating statistical distributioncharacteristics of physical parameters of a semiconductor device usingdata obtained through an analysis of correlation between actuallymeasured electrical characteristic data are provided. The methodsinclude: manufacturing a plurality of semiconductor device chips, eachhaving a plurality of transistors; preparing electrical characteristicdata by measuring electrical characteristics of the plurality oftransistors included in the plurality of chips; extracting an inter-chipdistribution characteristic and an intra-chip distributioncharacteristic of the electrical characteristics by analyzing theelectrical characteristic data; generating random number data satisfyingthe extracted inter-chip and intra-chip distribution characteristics;and performing a simulation for extracting statistical distributioncharacteristic data of the physical parameters of the chips, based onthe random number data.

In some embodiments, measuring of the electrical characteristicscomprises measuring a characteristic selected from the group consistingof: a drain-source current, a threshold voltage, and an off-current ofthe transistor.

In some embodiments, the extracting of the inter-chip and intra-chipdistribution characteristics includes: extracting statisticalcharacteristics of the respective electrical characteristics; anddetermining a correlation coefficient between the electricalcharacteristics using the statistical characteristics. Herein, theextracting of the statistical characteristics of the respectivecharacteristics includes calculating means and standard deviations ofthe respective electrical characteristics.

In other embodiments, the correlation coefficient is obtained bysubstituting measured values, means and standard deviations of therespective electrical characteristics into an equation:

$\rho_{xy} = {\frac{\text{cov}\left( {X,Y} \right)}{\sigma_{X}\sigma_{Y}} = \frac{E\left\lbrack {\left( {X - \mu_{X}} \right)\left( {Y - \mu_{Y}} \right)} \right\rbrack}{\sigma_{X}\sigma_{Y}}}$

where the X and Y denote respective measured values of selecteddifferent electrical characteristics, σ_(X) and σ_(Y) denote respectivestandard deviations of selected different electrical characteristics,μ_(X) and μ_(Y) denote respective means of selected different electricalcharacteristics, and cov and E denote a covariance and an expectationvalue, respectively.

In other embodiments, the extracting of the intra-chip distributionincludes: partitioning the chip into a plurality of sub-regions;selecting a predetermined transistor disposed in the predeterminedsub-region as a reference transistor; calculating a distance between thereference transistor and a selected transistor; extracting statisticalcharacteristics of the respective electrical characteristics accordingto the distance between the reference transistor and the selectedtransistor; and determining a correlation coefficient between theelectrical characteristics according to the distance between thereference transistor and the selected transistor, using thedistance-dependent statistical characteristics of the electricalcharacteristics.

In still other embodiments, the distance-dependent correlationcoefficient between the electrical characteristics is selected as thecorrelation coefficient of the inter-chip distribution characteristicwhen the selected transistor and the reference transistor are includedin the same sub-region and wherein the distance-dependent correlationcoefficient between the electrical characteristics is calculatedaccording to the distance between the reference transistor and theselected transistor when the selected transistor and the referencetransistor are not included in the same sub-region.

In even other embodiments, the extracting of the intra-chip distributioncharacteristic includes: preparing a correlation matrix in which thedistance-dependent correlation coefficient between the electricalcharacteristics is expressed on the basis of the distance between theselected transistor and the reference transistor and a correlationbetween different electrical characteristics; and obtaining a relationequation for the electrical characteristics of the transistors byanalyzing the correlation matrix using a multivariate statisticalanalysis technique. Herein, the analyzing of the correlation matrixusing a multivariate statistical analysis technique can be performedusing a principal component analysis (PCA) technique.

In yet other embodiments, the methods further include performing thesimulation based on the statistical distribution characteristic data ofthe physical parameters of the chips, to estimate characteristics ofchips to be subsequently manufactured.

BRIEF DESCRIPTION OF THE FIGURES

The accompanying figures are included to provide a further understandingof the embodiments of the present invention, and are incorporated in andconstitute a part of this specification. The drawings illustrateexemplary embodiments of the present invention and, together with thedescription, serve to explain principles of the present invention. Inthe figures:

FIG. 1 is a flowchart illustrating a conventional method for estimatingcharacteristics of a product;

FIG. 2 is a flowchart illustrating a method for estimating statisticaldistribution characteristics of physical parameters of a semiconductordevice in accordance with embodiments of the present invention;

FIGS. 3A, 3B and 3C are graphs illustrating distribution characteristicsof drain-source current (I_(ds)), threshold voltage (V_(th)) andoff-current (I_(off)) of transistors, respectively;

FIG. 4 is a flowchart illustrating a method for estimating an inter-chipdistribution characteristic in accordance with embodiments of thepresent invention;

FIGS. 5A, 5B and 5C are graphs illustrating correlations among thresholdvoltage (V_(th)), drain-source current (I_(ds)), and off-current(I_(off)) of transistors, respectively;

FIGS. 6A and 6B are graphs illustrating data distributions correspondingto predetermined correlation coefficients, respectively;

FIG. 7 is a flowchart illustrating a method for estimating an intra-chipdistribution characteristics in accordance with embodiments of thepresent invention;

FIGS. 8A, 8B and 8C are schematic views illustrating a method foranalyzing a position dependency of electrical characteristics inaccordance with an embodiment of the present invention; and

FIGS. 9A, 9B and 9C are graphs illustrating results of a method forestimating statistical distribution characteristics in accordance withan embodiment of the present invention.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

Preferred embodiments of the present invention will be described belowin more detail with reference to the accompanying drawings. The presentinvention may, however, be embodied in different forms and should not beconstrued as limited to the embodiments set forth herein. Rather, theseembodiments are provided so that this disclosure will be thorough andcomplete.

Exemplary embodiments of the present invention will now be described inconjunction with the accompanying drawings.

FIG. 2 is a flowchart illustrating a method for estimating statisticaldistribution characteristics of physical parameters of a semiconductordevice in accordance with embodiments of the present invention.

Referring to FIG. 2, a plurality of chips having a plurality oftransistors are manufactured. The chips may be formed on a single wafer,or on multiple, different, wafers. Subsequently, the electricalcharacteristics of the transistors included in the chips are measured(100), and thereafter, electrical characteristic data 110 are prepared.According to embodiments of the present invention, the electricalcharacteristic data 110 are used for analyzing the variation ofmanufacturing process of the chips and a correlation between theelectrical characteristics of the transistors.

In accordance with an embodiment of the present invention, theelectrical characteristics can include, for example, a drain-sourcecurrent (I_(ds)), a threshold voltage (V_(th)), an off-current(I_(off)), and the like. FIGS. 3A, 3B and 3C are graphs illustratingdistribution characteristics of drain-source current (I_(ds)), thresholdvoltage (V_(th)) and off-current (I_(off)) of transistors, respectively.However, the electrical characteristics are not limited to theseexemplified parameters, and thus, other electrical characteristics oftransistors such as a breakdown voltage of a gate insulating layer, abreakdown voltage of source/drain junction region, a punchthroughvoltage, or the like, may be selected as the electrical characteristicsfor which electrical characteristic data are prepared for analysis.

Through statistical analysis based on the electrical characteristic data110, there are extracted an inter-chip distribution characteristic data140 and an intra-chip distribution characteristic data 150. Herein, theinter-chip distribution characteristic data 140 contain statisticaldistribution characteristic data of electrical characteristics oftransistors included in different chips, whereas the intra-chipdistribution characteristic data 150 contain statistical distributioncharacteristic data of electrical characteristics of transistorsincluded in the same chip according to positions thereof. According tothe present invention, these inter-chip and intra-chip distributioncharacteristic data 140 and 150 contain data for the correlation betweenthe electrical characteristics. In addition, the inter-chip andintra-chip distribution characteristic data 140 and 150 may be obtainedthrough an inter-chip distribution characteristic analysis 120 and anintra-chip distribution characteristic analysis 130, respectively, whichwill be more fully illustrated herein with reference to FIGS. 4 and 7.

Thereafter, a plurality of random numbers are generated 160 such thatthey satisfy the inter-chip and intra-chip distribution characteristicdata 140 and 150. A set of the generated random numbers constitutesrandom number data 170 used in an analysis for extracting thestatistical characteristic data of physical parameters. That is, throughthe analysis 180 based on the random number data 170, there areextracted statistical characteristic data 190 of physical parameters(hereinafter, referred to as physical characteristic data 190) caused byvariations in the manufacturing process of the chip. Herein, theanalysis 180 for extracting the physical characteristic data 190 cancomprise, for example, a simulation using the random number data.

The physical characteristic data 190 contain expected statistical datafor physical/structural characteristics of the transistor such as achannel length, a thickness of a gate insulating layer, a channelconcentration, or the like. For example, in the case of mass productionof the chips, the physical characteristic data 190 may contain thestatistical data, i.e., mean, standard deviation, variance, etc, whichwill be expected with regard to the variation of the channel length.However, the physical characteristic data 190 are not limited to theexemplified ones, and thus it may include other physical/structuralcharacteristics of the transistor such as impurity concentrations ofgate electrode/source/drain, a channel width, depths of source/drain, adoping profile of the channel, and the like.

Following this, a simulation 200 on the basis of the physicalcharacteristic data 190 is performed so as to extract statisticaldistribution characteristic data of the chip quality 210, which is to beexpected during mass production of the chips. The chip quality cancontain, for example, operational characteristics of the chip such asaccess time and delay time, as well as DC properties and AC properties.The simulation for estimating the chip quality on the basis of thesephysical characteristic data 190 can be performed using a simulationtool such as a well-known SPICE or the like. However, according toembodiments of the present invention, since the physical characteristicdata 190 are obtained from measured data of electrical characteristicsof the manufactured chips, the distribution characteristic is differentfrom that provided in the conventional approach in that it is notassumed to be a normal distribution, but rather, it is actuallymeasured. In addition, according to embodiments of the presentinvention, the physical characteristic data 190 has a more increasedconformity with reality than that of the conventional approach, becausethey are obtained on the basis of information with regard to thecorrelation between the measured electrical characteristic data 110.

FIG. 4 is a flowchart illustrating a method for estimating an inter-chipdistribution characteristic in accordance with embodiments of thepresent invention.

Referring to FIG. 4, as described above, the electrical characteristicdata 110 contain a plurality of data for the electrical characteristicsmeasured from the transistors included in the manufactured chips. Theestimation of the inter-chip distribution characteristic according tothe present invention, includes: analyzing the electrical characteristicdata 120; obtaining respective means and standard deviations for theelectrical characteristics such as drain-source current (I_(ds)),threshold voltage (V_(th)) and off-current (I_(off)) 141; and obtainingthe correlation coefficient 142 between the respective electricalcharacteristics using a following equation 145.

$\begin{matrix}{\rho_{xy} = {\frac{\text{cov}\left( {X,Y} \right)}{\sigma_{X}\sigma_{Y}} = \frac{E\left\lbrack {\left( {X - \mu_{X}} \right)\left( {Y - \mu_{Y}} \right)} \right\rbrack}{\sigma_{X}\sigma_{Y}}}} & \left\lbrack {{Eq}.\mspace{14mu} 1} \right\rbrack\end{matrix}$

where X and Y denote respective measured values of selected differentelectrical characteristics, σ_(X) and σ_(Y) denote respective standarddeviations of the selected different electrical characteristics, μ_(X)and μ_(Y) denote respective means of the selected different electricalcharacteristics, and cov and E denote a covariance and an expectationvalue, respectively.

FIGS. 5A, 5B and 5C are graphs illustrating correlations among thresholdvoltage (V_(th)), drain-source current (I_(ds)), and off-current(I_(off)) of transistors, respectively. Specifically, axes of abscissaand ordinate of FIG. 5A represent the threshold voltages (V_(th)) andthe drain-source currents (I_(ds)), respectively, and axes of abscissaand ordinate of FIG. 5B represent the drain-source currents (I_(ds)),and the off-currents (I_(off)), respectively. In addition, axes ofabscissa and ordinate of FIG. 5C represent the threshold voltages(V_(th)) and the off-currents (I_(off)), respectively.

Referring to FIGS. 5A, 5B and 5C, it is understood that the thresholdvoltage (V_(th)), drain-source current (I_(ds)) and off-current(I_(off)) of the transistors are not independent from one another butrather, they have predetermined correlations. If there is no correlationamong them, points marked in the graphs would not demonstrateanisotropy. However, as illustrated in FIGS. 5A through 5C, it isconstrued that the electrical characteristics have considerablecorrelations because the points marked in the graphs are distributesalong specific identifiable profiles.

This correlation may be quantatively expressed as a correlationcoefficient, which may be obtained through Eq. 1. For example, in thegraph of FIG. 5A illustrating the correlation between the thresholdvoltages (V_(th)) and the drain-source currents (I_(ds)), thecorrelation coefficient ρ_(xy) is about −0.7 supposing that thevariables X and Y of Eq. 1 are the drain-source currents (I_(ds)) andthe threshold voltages (V_(th)), respectively. The obtained correlationcoefficients can be used in modeling the relation between the electricalcharacteristics. FIGS. 6A and 6B are graphs illustrating datadistributions when respective correlation coefficients are −0.94 and0.17, respectively.

FIG. 7 is a flowchart illustrating a method for estimating intra-chipdistribution characteristics in accordance with embodiments of thepresent invention.

Referring to FIG. 7, in accordance with the estimation of the intra-chipdistribution characteristic in accordance with embodiments of thepresent invention, electrical data 110 according to positions ofselected transistors are analyzed 130. Subsequently, there are obtainedmeans and standard deviations for respective electrical characteristics,e.g., the threshold voltage (V_(th)), drain-source current (I_(ds)) andoff-current (I_(off)) 151. Thereafter, the obtained means and standarddeviations are substituted into Eq. 2 below, and the correlationcoefficient 152 between the electrical characteristics according to theposition of the selected transistor is then obtained 155.

$\begin{matrix}{\rho_{xy}^{ij} = {\frac{\text{cov}\left( {X^{ij},Y^{ij}} \right)}{\sigma_{X}^{ij}\sigma_{Y}^{ij}} = \frac{E\left\lbrack {\left( {X^{ij} - \mu_{x}^{ij}} \right)\left( {Y^{ij} - \mu_{y}^{ij}} \right)} \right\rbrack}{\sigma_{X}^{ij}\sigma_{Y}^{ij}}}} & \left\lbrack {{Eq}.\mspace{14mu} 2} \right\rbrack\end{matrix}$

where the superscripts i and j are indexes for distinguishing respectiveselected transistors, X^(ij) and Y^(ij) denote respective measuredvalues of selected different electrical characteristics of the selectedtransistors, σ_(X) ^(ij) and σ_(Y) ^(ij) denote respective standarddeviations of selected different electrical characteristics of theselected transistor, and μ_(x) ^(ij) and μ_(y) ^(ij) denote respectivemeans of selected different electrical characteristics of the selectedtransistor.

FIGS. 8A, 8B and 8C are schematic views illustrating a method foranalyzing a position dependency of electrical characteristics inaccordance with one embodiment of the present invention.

Referring to FIGS. 8A, 8B and 8C, in order to increase the efficiency ofanalysis, the intra-chip distribution characteristic may be analyzed fortwo cases of which one case is that a transistor Tr₁ is spaced apartfrom a predetermined reference transistor Tr₀ by a separation distancewhich is smaller than a predetermined reference length D0, and anothercase is that transistors TR₂ and TR₃ are spaced apart from thepredetermined reference transistor Tr₀ by separation distances which aregreater than the predetermined reference length D0.

In detail, when the space D1 between the reference transistor and acomparison transistor is less than several, to several tens, ofmicrometers, it is construed that these two transistors are manufacturedunder nearly the same conditions. Therefore, the difference between theelectrical characteristics of the transistors can be estimated to berandom in this case. In other words, the distance dependency of theelectrical characteristic of the transistor may be negligible in thiscase. Herein, the reference distance D0 may be set to a predetermineddistance within which the distance dependency can be neglected, and itmay decrease/increase in consideration of the efficiency of analysis. Inthis case, i.e., D1<D0, the statistical distribution characteristic maybe analyzed based on the method for estimating the inter-chipdistribution characteristic, i.e., Eq. 1, as illustrated above withreference to FIG. 4.

However, when the distance between the reference transistor and thecomparison transistor is greater than the reference distance, thedifference between the electrical characteristics of the transistors isdependent upon the distance therebetween. Specifically, supposing thatthe distances between each of second and third comparison transistorsTr₂ and TR₃ and the reference transistor Tr₀ are D2 and D3,respectively, and an inequality condition is D3>D2>D0, the difference ofthe electrical characteristic between the second comparison transistorTr₂ and the reference transistor Tr₀ may be greater than the differenceof the electrical characteristic between the first comparison transistorTr₁ and the reference transistor Tr₀. Likewise, the difference of theelectrical characteristic between the third comparison transistor Tr₃and the reference transistor Tr₀ may be greater than the difference ofthe electrical characteristic between the second comparison transistorTr₂ and the reference transistor Tr₀. That is, as the distance betweenthe comparison transistor and the reference transistor increases, thecorrelation of the electrical characteristic between the referencetransistor Tr₀ and the comparison transistor is reduced.

The correlation having the distance dependency may be quantativelyexpressed as the correlation coefficient which can be obtained from Eq.2. FIG. 8B is a graph illustratively showing the distance dependency ofthe correlation coefficient. The obtaining of the correlation having thedistance dependency includes steps of preparing the electricalcharacteristic data 110 such that they contain data for positions ofmeasured transistors, and subsequently, obtaining the distance betweenthe measured transistor and the reference transistor using the positiondata. Subsequently, after analyzing the electrical characteristic data110 and obtaining the mean and the standard deviation according to theposition or distance of the measured transistor, these are substitutedinto Eq. 2 to thereby obtain a correlation coefficient. Referring toFIG. 8B, as described above, the correlation coefficient is a constant,i.e., ρ_(m), if D<D0, whereas the correlation coefficient, i.e., ρ(D),decreases as the distance D between the measured transistor and thereference transistor increases, if D>D0.

Meanwhile, in consideration of the efficiency of analysis, a singlesemiconductor chip can be partitioned into a plurality of sub-regions asillustrated in FIG. 8C. Each of the sub-regions may take the form of arectangle of which lengths of transverse and longitudinal sides are Land H, respectively. Herein, the lengths of transverse and longitudinalsides are determined on the basis of the aforementioned referencelength. According to one embodiment of the present invention, thelengths of transverse and longitudinal sides of the sub-region may beset to be equal to the reference length D0, i.e., L=D0 and H=D0. In thiscase, when the comparison transistor and the reference transistor areincluded in the same sub-region, the correlation coefficienttherebetween is determined as the correlation coefficient of theinter-chip distribution characteristic, i.e., the constant correlationcoefficient ρ_(m). On the contrary, when the comparison transistor andthe reference transistor are included in the different sub-regions, thecorrelation coefficient therebetween is determined using the correlationcoefficient function having the distance dependency as illustrated inFIG. 8B, after obtaining the distance between centers of the respectivesub-regions.

The obtained correlation coefficients may constitute a correlationmatrix for the electrical characteristics of all the transistorsincluded in the semiconductor chip. The correlation matrix is preparedsuch that it represents the correlation coefficient between theelectrical characteristics according to the distance, based on thedistance between the selected transistor and the reference transistorand the correlation between different electrical characteristics.Subsequently, the correlation matrix is analyzed using multivariatestatistical analysis technique so as to obtain the relation between theelectrical characteristics of the transistors. At this time, it ispreferable to make use of a principal component analysis (PCA) as themultivariate statistical analysis technique. After generating randomnumbers satisfying the inter-chip and intra-chip distributioncharacteristic data 140 and 150 using the relation, they are substitutedinto equations 3, 4 and 5 below, thereby obtaining distribution valuesof the respective electrical characteristics. Herein, the distributionvalues of the electrical characteristics are used as input data duringan analysis procedure 180 for extracting statistical characteristic data190 of the physical parameters.I _(ds)=μ(I _(ds))+σ(I _(ds))·R ₁  [Eq. 3]V _(th)=μ(V _(th))+σ(V _(th))·R ₂  [Eq. 4]I _(off)=μ(I _(off))+σ(I _(off))·R ₃  [Eq. 5]

FIGS. 9A, 9B and 9C are graphs illustrating results of a method forestimating statistical distribution characteristics in accordance withan embodiment of the present invention. In detail, FIGS. 9A, 9B and 9Care graphs that compare the statistical distribution characteristic data210 of the chip quality estimated through the disclosed embodiments ofthe present invention using actually measured electrical characteristicdata of the semiconductor chips. In addition, FIGS. 9A, 9B and 9Cillustrate results of drain-source current (I_(ds)), threshold voltage(V_(th)) and off-current (I_(off)) for respective transistors. Referringto FIGS. 9A, 9B and 9C, it may be understood that the statisticaldistribution characteristic data 210 of the chip quality estimatedthrough the results of the disclosed embodiments correlate considerablywith the actually measured electrical characteristic data of thesemiconductor chips.

The estimation for the characteristics of chip quality according to theembodiments of the present invention is obtained through correlationanalysis of electrical characteristic data having position data as wellas actual measured data. Specifically, the input data for a simulationof estimating the characteristics of the chip quality are prepared onthe basis of the actually-measured electrical characteristic data andthe random number data satisfying the correlation therebetween.Accordingly, the input data of the simulation has more increasedconformity with reality in comparison with that of the conventionaltechniques, and as a result, a more accurate estimation of thecharacteristic of chip quality can be achieved.

In addition, since the actually-measured electrical characteristics canbe electrically measured with ease, the time for measurement andanalysis is reduced. Such increased accuracy for estimation andreduction of time for measurement/analysis renders rapid productdevelopment period and reduced time-to-market more likely.

While embodiments of the invention have been particularly shown anddescribed above, it will be understood by those skilled in the art thatvarious changes in form and detail may be made herein without departingfrom the spirit and scope of the invention as defined by the appendedclaims.

1. A method for estimating statistical distribution characteristics ofphysical parameters of a semiconductor device, the method comprising:manufacturing a plurality of semiconductor device chips, each includinga plurality of transistors; measuring electrical characteristics of theplurality of transistors included in the plurality of actuallymanufactured chips so as to prepare actually measured electricalcharacteristic data; analyzing the actually measured electricalcharacteristic data so as to extract an inter-chip distributioncharacteristic and an intra-chip distribution characteristic of theelectrical characteristics; generating random number data satisfying theextracted inter-chip and intra-chip distribution characteristics; andperforming a simulation based on the random number data to extractstatistical distribution characteristic data of the physical parametersof the chips.
 2. The method of claim 1, wherein the measuring of theelectrical characteristics comprises measuring a characteristic selectedfrom the group consisting of: a drain-source current, a thresholdvoltage, and an off-current of the transistor.
 3. The method of claim 1,wherein the extracting of the inter-chip and intra-chip distributioncharacteristics comprises: extracting statistical characteristics of therespective electrical characteristics; and determining a correlationcoefficient between the electrical characteristics using the statisticalcharacteristics.
 4. The method of claim 3, wherein the extracting of thestatistical characteristics of the respective characteristics comprisesobtaining means and standard deviations of the respective electricalcharacteristics.
 5. The method of claim 4, wherein the correlationcoefficient is obtained by substituting measured values, means andstandard deviations of the respective electrical characteristics into anequation:$\rho_{xy} = {\frac{\text{cov}\left( {X,Y} \right)}{\sigma_{X}\sigma_{Y}} = \frac{E\left\lbrack {\left( {X - \mu_{X}} \right)\left( {Y - \mu_{Y}} \right)} \right\rbrack}{\sigma_{X}\sigma_{Y}}}$where the X and Y denote respective measured values of selecteddifferent electrical characteristics, σ_(X) and σ_(Y) denote respectivestandard deviations of selected different electrical characteristics,μ_(X) and μ_(Y) denote respective means of selected different electricalcharacteristics, and cov and E denote a covariance and an expectationvalue, respectively.
 6. The method of claim 3, wherein the extracting ofthe intra-chip distribution characteristic comprises: partitioning thechip into a plurality of sub-regions; selecting a predeterminedtransistor disposed in the predetermined sub-region as a referencetransistor; obtaining a distance between the reference transistor and aselected transistor; extracting statistical characteristics of therespective electrical characteristics according to the distance betweenthe reference transistor and the selected transistor; and determining adistance-dependent correlation coefficient between the electricalcharacteristics using the distance-dependent statistical characteristicsof the electrical characteristics.
 7. The method of claim 6, wherein thedistance-dependent correlation coefficient between the electricalcharacteristics is selected as the correlation coefficient of theinter-chip distribution characteristic when the selected transistor andthe reference transistor are included in the same sub-region, andwherein the distance-dependent correlation coefficient between theelectrical characteristics is obtained according to the distance betweenthe reference transistor and the selected transistor when the selectedtransistor and the reference transistor are not included in the samesub-region.
 8. The method of claim 7, wherein the extracting of theintra-chip distribution characteristic comprises: preparing acorrelation matrix in which the distance-dependent correlationcoefficient between the electrical characteristics is expressed on thebasis of the distance between the selected transistor and the referencetransistor and a correlation between different electricalcharacteristics; and obtaining a relation equation for the electricalcharacteristics of the transistors by analyzing the correlation matrixusing a multivariate statistical analysis technique.
 9. The method ofclaim 8, wherein the analyzing of the correlation matrix using amultivariate statistical analysis technique is performed using a PCA(principal component analysis) technique.
 10. The method of claim 1,further comprising: performing the simulation based on the statisticaldistribution characteristic data of the physical parameters of the chipsto estimate characteristics of chips, to be subsequently manufactured.11. A method for estimating statistical distribution characteristics ofphysical parameters of a semiconductor device, the method comprising:manufacturing a plurality of semiconductor device chips, each includinga plurality of transistors; measuring electrical characteristics of theplurality of transistors included in the plurality of chips so as toprepare electrical characteristic data; analyzing the electricalcharacteristic data so as to extract an inter-chip distributioncharacteristic and an intra-chip distribution characteristic of theelectrical characteristics; generating random number data satisfying theextracted inter-chip and intra-chip distribution characteristics; andperforming a simulation based on the random number data to extractstatistical distribution characteristic data of the physical parametersof the chips, wherein the extracting of the inter-chip and intra-chipdistribution characteristics comprises: extracting statisticalcharacteristics of the respective electrical characteristics; anddetermining a correlation coefficient between the electricalcharacteristics using the statistical characteristics, wherein theextracting of the statistical characteristics of the respectivecharacteristics comprises obtaining means and standard deviations of therespective electrical characteristics, and wherein the correlationcoefficient is obtained by substituting measured values, means andstandard deviations of the respective electrical characteristics into anequation:$\rho_{xy} = {\frac{\text{cov}\left( {X,Y} \right)}{\sigma_{X}\sigma_{Y}} = \frac{E\left\lbrack {\left( {X - \mu_{X}} \right)\left( {Y - \mu_{Y}} \right)} \right\rbrack}{\sigma_{X}\sigma_{Y}}}$where the X and Y denote respective measured values of selecteddifferent electrical characteristics, σ_(X) and σ_(Y) denote respectivestandard deviations of selected different electrical characteristics,μ_(X) and μ_(Y) denote respective means of selected different electricalcharacteristics, and cov and B denote a covariance and an expectationvalue, respectively.
 12. The method of claim 11, wherein the measuringof the electrical characteristics comprises measuring a characteristicselected from the group consisting of: a drain-source current, athreshold voltage, and an off-current of the transistor.
 13. The methodof claim 11, wherein the extracting of the intra-chip distributioncharacteristic comprises: partitioning the chip into a plurality ofsub-regions; selecting a predetermined transistor disposed in thepredetermined sub-region as a reference transistor; obtaining a distancebetween the reference transistor and a selected transistor; extractingstatistical characteristics of the respective electrical characteristicsaccording to the distance between the reference transistor and theselected transistor; and determining a distance-dependent correlationcoefficient between the electrical characteristics using thedistance-dependent statistical characteristics of the electricalcharacteristics.
 14. The method of claim 13, wherein thedistance-dependent correlation coefficient between the electricalcharacteristics is selected as the correlation coefficient of theinter-chip distribution characteristic when the selected transistor andthe reference transistor are included in the same sub-region, andwherein the distance-dependent correlation coefficient between theelectrical characteristics is obtained according to the distance betweenthe reference transistor and the selected transistor when the selectedtransistor and the reference transistor are not included in the samesub-region.
 15. The method of claim 14, wherein the extracting of theintra-chip distribution characteristic comprises: preparing acorrelation matrix in which the distance-dependent correlationcoefficient between the electrical characteristics is expressed on thebasis of the distance between the selected transistor and the referencetransistor and a correlation between different electricalcharacteristics; and obtaining a relation equation for the electricalcharacteristics of the transistors by analyzing the correlation matrixusing a multivariate statistical analysis technique.
 16. The method ofclaim 15, wherein the analyzing of the correlation matrix using amultivariate statistical analysis technique is performed using a PCA(principal component analysis) technique.
 17. The method of claim 11,further comprising: performing the simulation based on the statisticaldistribution characteristic data of the physical parameters of the chipsto estimate characteristics of chips, to be subsequently manufactured.18. A method for estimating statistical distribution characteristics ofphysical parameters of a semiconductor device, the method comprising:manufacturing a plurality of semiconductor device chips, each includinga plurality of transistors; measuring electrical characteristics of theplurality of transistors included in the plurality of chips so as toprepare electrical characteristic data; analyzing the electricalcharacteristic data so as to extract an inter-chip distributioncharacteristic and an intra-chip distribution characteristic of theelectrical characteristics; generating random number data satisfying theextracted inter-chip and intra-chip distribution characteristics; andperforming a simulation based on the random number data to extractstatistical distribution characteristic data of the physical parametersof the chips, wherein the extracting of the inter-chip and intra-chipdistribution characteristics comprises: extracting statisticalcharacteristics of the respective electrical characteristics; anddetermining a correlation coefficient between the electricalcharacteristics using the statistical characteristics, wherein theextracting of the intra-chip distribution characteristic comprises:partitioning the chip into a plurality of sub-regions; selecting apredetermined transistor disposed in the predetermined sub-region as areference transistor; obtaining a distance between the referencetransistor and a selected transistor; extracting statisticalcharacteristics of the respective electrical characteristics accordingto the distance between the reference transistor and the selectedtransistor; and determining a distance-dependent correlation coefficientbetween the electrical characteristics using the distance-dependentstatistical characteristics of the electrical characteristics.
 19. Themethod of claim 18, wherein the distance-dependent correlationcoefficient between the electrical characteristics is selected as thecorrelation coefficient of the inter-chip distribution characteristicwhen the selected transistor and the reference transistor are includedin the same sub-region, and wherein the distance-dependent correlationcoefficient between the electrical characteristics is obtained accordingto the distance between the reference transistor and the selectedtransistor when the selected transistor and the reference transistor arenot included in the same sub-region.
 20. The method of claim 19, whereinthe extracting of the intra-chip distribution characteristic comprises:preparing a correlation matrix in which the distance-dependentcorrelation coefficient between the electrical characteristics isexpressed on the basis of the distance between the selected transistorand the reference transistor and a correlation between differentelectrical characteristics; and obtaining a relation equation for theelectrical characteristics of the transistors by analyzing thecorrelation matrix using a multivariate statistical analysis technique,wherein the analyzing of the correlation matrix using a multivariatestatistical analysis technique is performed using a PCA (principalcomponent analysis) technique.